Combinatorial Group Testing Problems on Various Models Speaker: - PowerPoint PPT Presentation

Combinatorial Group Testing Problems on Various Models Speaker: Huilan Chang Advisor: Hung-Lin Fu National Chiao Tung University Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Group Testing Problem Classical group testing: Origination: blood testing in World War II Venkatesh Saligrama at Boston University Rosenblatt and Dorfman (1940s) suggested to pool the blood 1 / 2 6 samples. Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Group Testing Problem A set N of n clones, each of which is either positive or negative, is given. Goal: identify all positive ones by as few number of group tests as possible. A group test is an experiment on a subset of clones. positive 4 T 3 1 5 positive clone 3 negative clone 2 6 T 2 positive T 1 negative 2 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Group Testing Problem In combinatorial group testing , the number of positive items is usually assumed at most d ( << n ) . 3 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Preliminaries on Algorithms A nonadaptive algorithm specifies all tests simultaneously. Matrix representation: clones 1 2 3 4 5 6 outcomes 1 1 1 0 0 0 1 tests 1 0 0 1 1 0 0 (pools) 0 1 0 1 0 1 0 0 0 1 0 1 1 1 4 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Preliminaries on Algorithms 1 2 3 4 5 6 outcomes 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 {1,2}\{1,4}={2} 5 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Preliminaries on Algorithms 1 2 3 4 5 6 outcomes 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 {1,2}\{1,4}={2} A binary matrix is d -disjunct if for any d + 1 columns C 0 , C 1 , · · · , C d , d � � � � � ≥ 1 . � C 0 \ � C i � � � � i = 1 � τ 0 ( C ) > 0 for a negative clone C , Then τ 0 ( P ) = 0 6 / 2 6 for a positive clone P . Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Various Group Testing Models Inhibitor Model (Farach et al. , 1997) Complex Model (Torney, 1999) Inhibitor Complex Model (Chang et al. , 2010a) Graph Reconstruction 7 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Model An inhibitor is a third type of clones whose existence may cancel the effect of positive clones (Farach et al. , 1997). Application: • In drug discovery, certain compound can block the detection of potent compounds (Xie et al. 2001). • Enzyme inhibitors are molecules that interact in some way with the enzyme to prevent it from working normally. 8 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Complex Model A subset of clones, called a complex , can induce a positive or negative effect (Torney, 1999). General setting: no complex contains another. Application: Protein-to-protein interaction (Lappe and Holm, 2004): http://www.mdc-berlin.de 9 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Inhibitor Complex Model (Chang et al. , 2010a): besides the positive complexes and negative complexes, an inhibitor is a third type of complexes. negative complex 2 1 3 positive complex 4 5 2 3 4 3 4 5 inhibitor positive outcome negative outcome 1 0 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Inhibitor Complex Model (Chang et al. , 2010a): besides the positive complexes and negative complexes, an inhibitor is a third type of complexes. negative complex 2 1 3 positive complex 4 5 2 3 4 3 4 5 inhibitor positive outcome negative outcome Consider the setting: a group test is positive only when it contains a positive complex and no inhibitor. Combinatorial assumption: there are at most d positive complexes and at most h inhibitors, and each complex contains 1 1 / 2 6 at most r clones. Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model A binary matrix is ( d , r ; z ] -disjunct if for any r + d columns C 1 , · · · , C r + d , r r + d � � � � � � C i \ C i � ≥ z . � � � � i = 1 i = r + 1 � Such matrics are closely related to MDS codes (D’yachkov et al., 2002) and separating hash families (Stinson et al. 2004). X * * 1 2 3 4 5 6 outcomes negative negative or inhibitor tests z 1 1 0 0 0 0 0 positive (pools) For a negative or inhibitory complex X , 1 2 / 2 6 τ 0 ( X ) ≥ z . Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model A matrix is ( h , r ; y ] -inclusive if for any r + h columns C 1 , · · · , C r + h , r r + h � � � � � � � C i ) C i ) � ≤ y . � ( ( � � � � i = 1 i = r + 1 h P * * 1 2 3 4 5 6 outcomes inhibitor ests 1 1 1 0 0 0 0 positive y ools) 1 1 1 1 0 0 0 For a positive complex P with | P | = r , τ 0 ( P ) ≤ y . ( P appears in at 1 3 / 2 6 most y negative pools.) Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Theorem (Chang et al., 2010b) A ( d , r ; z ] -disjunct and ( h , r ; y ] -inclusive matrix with z > y can solve the general inhibitor complex model when each positive complex contains exactly r clones. Decoding procedure: { X : τ 0 ( X ) ≤ y } is the set of positive complexes. 1 4 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Our results on the inhibitor complex model (Chang et al, 2010a, 2010b): (A) ( H , d + h ; 2 e + 1 ) -disjunct, H is the set of complexes. (B) ( d + h , r ; 2 e + 1 ] -disjunct. (C) ( d , r ; z ] -disjunct and ( h , r ; y ] -inclusive with z − e > y + e . (A) ⇐ (B) ⇐ (C). 1 5 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Our results on the inhibitor complex model (Chang et al, 2010a, 2010b): (A) ( H , d + h ; 2 e + 1 ) -disjunct, H is the set of complexes. (B) ( d + h , r ; 2 e + 1 ] -disjunct. (C) ( d , r ; z ] -disjunct and ( h , r ; y ] -inclusive with z − e > y + e . (A) ⇐ (B) ⇐ (C). Design Decoding complexity Testing performance O (( 2 ( d + h ) r lg n � | H |− 1 lg (( d + h ) r lg n )) r + 1 ) O ( ( hr ) t | H | ) � (A) h (Gao et al. ,2006) � n − r O ( c ( d , h , r ) lg n ) O ( ( h + r ) t | H | ) � (B) h O ( rt | H | ) (C) (Chang et al. 2010b) r = 1 : O (( d + h ) 2 lg n ) 1 6 / 2 6 (Hwang and S´ os, 1987) r > 1 : T -design, direct constructions Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Classification problem on inhibitor complex model: classify all complexes by group tests. Motivation: Enzyme inhibitors are molecules that bind to enzymes and decrease their activity. Since blocking an enzyme’s activity can kill a pathogen or correct a metabolic imbalance, many drugs are enzyme inhibitors. 1 7 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Inhibitor Complex Model Theorem (Chang et al., 2010b) A ( d + h , 2 r ; 1 ] -disjunct matrix can classify all complexes under the inhibitor complex model. Decoding procedure: Identify inhibitors first. • { X : τ 1 ( X ) = 0 } ≡ I is the set of inhibitors. Then distinguish positive from negative. 0 ( X ) = 0 } is the set of positive complexes. • { X ∈ H \ I : τ I X 1 8 / 2 6 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Graph Reconstruction Given a family G of labeled graphs on [ n ] , a hidden graph G is known belonging to G . • Example: G = { Hamiltonian cycles on [ 4 ] } = 1 1 1 4 2 4 3 3 2 3 2 4 The main task is to reconstruct G by asking queries as few as possible. A query Q ( S ) for S ⊆ [ n ] is “Does S induce at least one edge of G ?” . 1 9 / 2 6 • Example: Q ( { 1 , 3 } ) = 0 . Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Graph Reconstruction Given a family G of labeled graphs on [ n ] , a hidden graph G is known belonging to G . • Example: G = { Hamiltonian cycles on [ 4 ] } = 1 1 1 4 2 4 3 3 2 3 2 4 The main task is to reconstruct G by asking queries as few as possible. A query Q ( S ) for S ⊆ [ n ] is “Does S induce at least one edge of G ?” . 2 0 / 2 6 • Example: Q ( { 1 , 3 } ) = 0 . Then G = 1234 . Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models